Question

A) Given that the life of Sunshine CD players is normally distributed with a mean of 4.1 years and a standard deviation of 1.3 years, find the probability that a CD player will break down during the guarantee period of three years. Calculate P(0 < X < 3), where X represents the lifespan of the CD player.

b) Determine the probability that a CD player will last between 2.8 and six years. Calculate P(2.8 < X < 6), where X represents the lifespan of the CD player.

c) Find the 70th percentile of the distribution for the time a CD player lasts. In other words, what is the value of X such that the probability that a CD player lasts less than X is 0.70?

Answer 1

To find the probability that a CD player will break down during the guarantee period of three years, we can use the z-score formula and the standard normal distribution table.

Step-by-step explanation:

To find the probability that a CD player will break down during the guarantee period of three years, we need to find the probability that the lifespan of the CD player is between 0 and 3 years. To do this, we can use the z-score formula and the standard normal distribution table.

First, we need to find the z-scores for 0 and 3 years using the formula:
z = (x - mean) / standard deviation

For 0 years:
z = (0 - 4.1) / 1.3 = -3.08

For 3 years:
z = (3 - 4.1) / 1.3 = -0.85

Next, we can use the standard normal distribution table to find the probabilities corresponding to these z-scores. The probability that the lifespan of the CD player is between 0 and 3 years is the difference between these probabilities:

P(0 < X < 3) = P(-3.08 < Z < -0.85)

Using the standard normal distribution table, we can find that P(-3.08 < Z < -0.85) is approximately 0.1967. Therefore, the probability that a CD player will break down during the guarantee period is approximately 0.1967.